Computing the coefficients of a recurrence formula for numerical integration by moments and modified moments

摘要

To evaluate the class of integrals $\int^1_{-1}e^{-\alpha x}f(x) dx$, where $\alpha \in \R^+$ and the function f(x) is known only approximately in a tabular form, we wish to use a Gaussian quadrature formula. Nodes and weights have to be computed using the family of monic orthogonal polynomials, with respect to the weight function $e^{-\alpha x}$, obtained through the three-term recurrence relation $P_{k+1}(x) = (x+B_{k+1})P_k(x)-C_{k+1}P_{k-1}(x)$.To guarantee a good precision, we must evaluate carefully the values for the coefficients $B_{k+1}$ and $C_{k+1}$. Such evaluations are made completely formally through a Mathematica program to obtain great precision.A comparison between various methods, starting from moments and modified moments, is shown. Numerical results are also presented.